진동의 기초 - shinhoent.co.kr · 부경대학교 기계공학부 1 Intelligent Mechanics Lab. 진동의 기초 Fundamentals of Vibration . 양보석 교수 . 부경대학교

Intelligent Mechanics Lab. 1

Fundamentals of Vibration

(2006, 1)


() : () () (), , , ,

( ): ) (pendulum) , (electron) , , rolling, (wave motion) Oscillation

( ): ( ) , , , : Hz kHz ( ) : 1 Hz 100 Hz Vibration

Intelligent Mechanics Lab. 3 1

(, system) : , (device) (ISO ) : , , ,

(analysis) : . .

System Input Output


(linear system)

() (Linear system) () (Non-linear system) () (Time-invariant system) () (Time-variant system)

(time-variant system) : , k(t)

( ) ( ) ( ) ( )t t t t+ + =xM C Kx x F

( ) ( ) ( ) )( ) (t t t tt+ + =Mx C xKx F

x

F F = k x, k = constant F = k(x) x, k = ax + bx2


(vibrating system) : , (mass), (stiffness), (damping)

1(1 degree of freedom system) : (multi-degree of freedom system) : 2 () (discrete system) : () (continuous system) :

() ()

1 ()

x q(x, t)

q1(t) q1(t) q2(t) q3(t)

3 DOF 1 DOF


Discrete model (1 DOF)

x F(t)

Inertia force distribution Over-estimated Over-estimated, better

Continuous model Discrete model (3 DOF)

Items Continuous model Discrete model

Equation of motion Partial differential equation Ordinary differential equation

Variables Time, geometry (x, y, z) Time only

Solution accuracy Exact solution Approximate solution

Solution type Analytical solution Numerical solution

Solution procedure Very complex, limited Easy, Computer used


(natural frequency) (natural mode)

(free vibrations) (forced vibrations) (self-excited vibrations)

(harmonic forces) (periodic forces) (impulsive forces) (random forces)

(mass) (stiffness) (damping)

(mass) : (inertia force) () (stiffness) : () x () F (k = F/x) (elastic restoring force) (damping) : (dissipation), (damping force): :


)( tFq Kq CqM =++

)(tF )(tq


HP Rotor LP Rotor

Turbine/Generator Rotor System


(sine wave) :

-180 0 180 360 540A

(f = 1 / T)

T

Angle (deg)

y

t

3

)2sin( += ftAy

(frequency) f :

(amplitude) A : ,

(phase) f :


(Unit) : Hz (Hertz) : 1 (cps ; cycle per second), f cpm (cycle per minute) : 1 , N n N = 60 f (cpm) (angular frequency) : , (omega) = 2f (rad/s) (period) : 1 () (Hz) T = 1 / f (s)

() : f


peak

(displacement), (velocity), (acceleration) , , ,

peak-to-peak rms

peak 0-to-peak(, p) : , peak-to-peak (, , p-p) : rms (root mean square, ) : , ,

, (= 0.707 peak)

(magnitude)


y1(t), y2(t) : deg, rad

(Phase)

y1(t)

y2(t)


: : mm/s, IPS(in/s)

:

: m/s2, g( G)

22 // dtyddtdva ==

dtdyv /=)sin( += tAy

)cos( += tAv

)sin(2 += tAa


Newton 2: Newton : m d 2x/dt2 = F Newton ()

m d 2x/dt2 = F = Fspring + Fdamping + Fexternal = kx c dx/dt + F(t)

maF =Fma =

)(tFkxxcxm =++

Equation of Motion


3

M ( )

kg ( )

kgm2

C (/)

Ns/m (/)

Nms/rad

K (/)

N/m (/)

Nm/rad


1

(F = 0, c =0),

0=+ kxxm

tXtx sin)( =

tXtx sin)( 2=

0sin)( 2 =+ tXkm

X 0 ( )

02 =+ km mk /=

, 2

,

(Natural frequency)

= n


m

k

x(t)

m

c k

x(t)

taetx =)(

+= + tjtjt eaeaetx

22 12

11)(

+= + ttt eaeaetx 12

11

22

)(

( ) tetaatx += 21)(

1

( ) ( ) 0mx t kx t+ =

1 2( )j t j tx t a e a e + = +

( ) tx t Ae=

( )( ) ( ) 0mx t kxcx t t+ + =


tjtj eaeatx + += 21)(

+= + tjtjt eaeaetx

22 12

11)(

( ) += tAtx sin)(

tAtAtx sincos)( 21 += ( )tAtAetx ddt sincos)( 21 +=

( ) += tAetx dt sin)(

211 aaA += ( ) jaaA 212 =

22

21 AAA += ( )211 /tan AA=

221

1jAAa +=

221

2jAAa =

.

1


Potential energy (PE)

Kinetic energy (KE)

Total energy (TE)

m

k

n = 2 fn = (elasticity / inertia)1/2

x(t) = X cos (n t + )

PE = k x2 = k X 2 cos2 (n t + )

KE = mv2 = m (dx/dt)2

= m n2 X 2 sin2 (n t + )

TE = P.E. + K.E. = k x2 + mv2

= k X 2 cos2 (n t + )

+ m n2 X 2 sin2 (n t + )

= k X 2 (cos2 n t + sin2 n t )

= k X 2 = mV 2


(natural frequency) ? () (m) (k)

() :

:

?

(resonance), (critical speed)

mkn /=

mkfn 2

1=

(rad/s)

(Hz)


m

k

x(t)

F(t)

m

c k

x(t)

F(t) tFtF drcos)( 0=

1

0( ) ( ) cos drmx t kx t F t+ = 0( ) ( ) ( ) cos drmx t cx t kx t F t+ + =

Harmonic excitation

Undamped Vibration Model Damped Vibration Model


tAx drp cos0=

tf

tx drdrn

p cos)( 22

0

=

mF

f 00 = mk

n =tAtAtx nnh cossin)( 21 +=

tf

tAtAtx drdrn

nn coscossin)( 22

021

++=

m

k

x(t)

F(t)

1

(Undamped Vibration Model)

0( ) ( ) cos drmx t kx t F t+ =

Forced response Free response


220

20 )0(drn

fAxx

+== 10 )0( Axv n==

tf

tf

xtv

tx drdrn

ndrn

nn

coscossin)( 220

220

00

+

+=

1

: 01.00 =x01.00 =v

ndr

0f

= 1 rad/s

= 0.1 N/kg

m m/s

= 2 rad/s

, 2 (n, dr) (2 )


(Beat)

000 == vx

( ))cos()cos()( 22 0 ttftx ndr

drn

=

+

= tt

ftx drndrn

drn 2sin

2sin

2)( 22

0

2 (n dr)

, ,

: 2 ( fn fdr) 1

2n dr n drf f f

= =


(resonance)

ttAtx drp sin)( 0=

ttf

tx drn

p sin

2)( 0=

ttf

tAtAtx drn

nn sin

2cossin)( 021 ++=

ttf

txtv

tx nrn

nnn

sin2

cossin)( 000 ++=

2 (n = dr)


mF

f 00 = mk

n =

m

c k

x(t)

F(t)

nmc

2

=

)cos()( 0 = tAtx drp

+= 22

1

2222

0 2tancos)2()(

)(drn

drndr

drndrn

p tf

tx

)cos()sin()( 0 ++= drd

t AtAetx

1

(Damped Vibration Model)

0( ) ( ) cos( ) drmx t kx t Fc t tx + + =

20( ) 2 ( ) ( ) cosn n drx t x t x t f t + + =

Forced response Free response


)cos()sin()( 0 ++= drd

t AtAetx

2222

00

)2()( drndrn

fA

+=

221 2tan

drn

drn

=

2220

20

0

0

)2()1(1

rrfA

FkA n

+== 2

1

12tan

rr

=

2220

0)()( drdr cmk

FA +

=2

1tandr

dr

mkc

=

1

Forced response/ Steady state response

Free response/ Transient response

Amplification factor (dynamic/static) Phase angle


:

:

tFkxxcxm sin0=++

)sin(0 = txx

22220

2220

0)/2(])/(1[

/)()( n

kFcmk

Fx +

=+

=

kmc

cc

r 2==

211

)/(1/2tan)(tan

n

n

mkc

=

=

(damping ratio)

,

n n

frf

= =

(frequency ratio)

Static response


() :

:

2220

0

)]/(2[])/(1[1

/nn ffffkF

xM+

==

21

)/(1)/(2tan

n

n

ffff

=

nff /


()

0 1 2 3 4 5

Frequency Ratio

1

3

5

7

9

0

2

4

6

8

10

Mag

nifie

r

0.00.05

0.10

0.15

0.250.3750.501.0

Dynamic Magnification Factor

Damping Ratio


0.0 1.0 2.0 3.0 4.0 5.0Frequency Ratio

0

30

60

90

120

150

180

Phas

e (d

eg)

0.0 0.050.10

0.150.250.375

0.501.0

Response Phase Delay

Damping Ratio


(resonance) ? f fn ( f = fn)

() (quality factor: Q factor)

2/1=Q


, (resonance) F , 1 X

Resonance (f = fn)


: 2 . : (system) M M , fn fn fe fn , Xe . Xe

( )


X B B+B , Xe Xe fn (viscous damper)

( )


, Xe -() , fn 2 ( f1n f2n ) M C B

( )

Secondary system

Main vibration system


Vibration Isolation


F (t) (foundation) ,

( / )

)(tFkxxcxm =++

222

2

)]/(2[])/(1[

)]/(2[1

nn

nT

ffff

ffFFTR

+

+==

)(tFkxxc T=+

XkicmF )( 2 ++=

XkicFT )( +=

:

:


(foundation)

,

( / )

0)()( =++ bb xxkxxcxm

bb kxxckxxcxm +=++

bXkicXkicm )()(2 +=++

222

2

)]/(2[])/(1[

)]/(2[1

nn

n

b ffff

ffXXTR

+

+==

tieXx = tibb eXx=


(Transmissibility)

,

1 .

, 1 .

.

222

2

)]/(2[])/(1[

)]/(2[1

nn

n

ffff

ffTR

+

+=

)/( nff 2

)/( nff 2


0 1 2 3 4 5Frequency Ratio

1

3

5

0

2

4

Tran

smiss

ibilit

y

0.00.05

0.10

0.15

0.25

0.3750.501.0

Vibration Transmissibility

Damping Ratio

2

(Transmissibility)


, 1

(resilient mounting design)

mounting , ()

3, 0.1

(Vibration Isolation)

2


(Dynamic Absorber)

Main mass

Absorber mass

F0 sin t


(Harmonic Analysis)

(periodic signal) :

: Fourier

: , , , ,


Diesel engine combustion chamber Pressure change

Gas turbine Shaft whirling

Ship propulsion shaft Torsional vibration


(Harmonic Analysis)

Joseph Fourier : " ." ,

....3sin 2sin sin ....3cos2coscos)(

321

3210

++++++++=

tbtbtbtatataatx

=T

dttxT

a00

)(1

=T

n dttntxTa

0 cos)(2

=T

n dttntxTb

0 sin)(2

:

(n = 1, 2, 3, )

(n = 1, 2, 3, ) Joseph Fourier (17681830)


n :

(harmonic coefficient)

Fourier series, harmonics

)cos(sincos nnnn tnctnbtna =+

, 22 nnn bac += )/(tan1

nnn ab=

nc n (nth order component)

nn ba ,

nnc ,( )

( )

?

!


(fundamental frequency) :

(high order) : ( 2, 3, 4 )

(order) : (1, 2, 3 )

: " (sine wave)

/ ? " sine (sine cosine ) , "


-4 -3 -2 -1 0 1 2 3 4-1.5

-1

-0.5

0

0.5

1

1.5

Time

Am

plitu

de

Fourier series Diagram

(1)

(rectangular wave)


(rectangular wave)

1st order

3rd order

5th order

7th order

9th order

11th order

13th order

Original

Sum up to each order

15th order

Original

(2)


Output torque of single cylinder engine

1st order

2nd order

3rd order

4th order

5th order

6th order

7th order

Original

Sum up to each order

8th order

Original

(3)


(continuous system)

(discrete system)

()

(analytical) (numerical)


(mode)

(vibration mode) ?

Mode, Normal Mode, Mode Shape, Mode Vector, Eigenvector, Elastic Curve, Relative Amplitude


(1 ) (1 degree of freedom system) 1 . (multi-DOF system) .

.

.

.

.

1st mode

2nd mode

3rd mode

(mode)

1 DOF 2 DOF 3 DOF

1 DOF 2 DOF

3 DOF

q1(t) q2(t) q3(t)

3 DOF

q1(t) q2(t)

2 DOF

q1(t)

1 DOF


(string)

:

:

/Tl

nn =

lxnxYn sin)( =

( l : , T : , : ) 1st Mode

2nd Mode3rd Mode

Node


3

:

x 1 x 2 x 3

m m m k k k

mk /4450.01 =

mk /2470.12 =

mk /8019.13 =

1

2

3


(2, 1) (1, 2) (2, 1)+(1, 2) (2, 1)-(1, 2)

(3, 1) (1, 3) (3, 1)+(1, 3) (3, 1)-(1, 3)

(1, 1) (2, 2) Plate


n (nth order) n (nth mode)

/

: 2 0


(modal analysis)

Excitation

Transducer


K C

M

F x

x F

t

F

x

x

x

t

t

t

?

?

Input (Force)

Output (Response) System


,


(exciting force)

(harmonics)

(natural frequency) (mode)

=K x F

( ) ( ) ( ) ( )t t t t+ + =M x Cx K x F


(Vibration assessment) :

:

(condition monitoring & diagnosis) (prediction & maintenance)

() :

(vibration quality)


Assignment

Self-study & Presentation Presentation material using Power Point File
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Documents

진동의 기초 - shinhoent.co.kr · 부경대학교 기계공학부 1 Intelligent Mechanics Lab. 진동의 기초 Fundamentals of Vibration . 양보석 교수 . 부경대학교